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Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $Y$ the functor $R \mapsto Y(R((h)))$ is a sheaf). Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?

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    $\begingroup$ Welcome new contributor. No, that morphism of Zariski sheaves is not surjective. There is a discussion of this in Serre's "Galois cohomology". $\endgroup$
    – Jason Starr
    Commented Apr 28, 2020 at 22:56
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    $\begingroup$ @JasonStarr thanks a lot! But could you be a little more specific? In which chapter can I find this discussion? $\endgroup$
    – Ekaterina Bogdanova
    Commented Apr 29, 2020 at 7:16
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    $\begingroup$ I do not have a copy of the book with me, but it should be prior to the long exact sequence of non-Abelian cohomology. For instance, if $R$ equals $\mathbb{C}[x,y]/\langle y^2-x^2(x-1)\rangle$, the $n$-torsion elements in the Picard group give counterexamples. $\endgroup$
    – Jason Starr
    Commented Apr 29, 2020 at 17:01
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    $\begingroup$ Dear @JasonStarr, I am confused. I've looked it up in the Serre`s book and have not find information concerning this question. To this point I am not sure that I formulated it clear enough. My question is the following. For any ring $R$ an element in $PGL(R((h)))$ defines a line bundle $L$ on $Spec(R((h)))$. I wonder whether it is true that there exists an open covering $Spec R = \cup Spec R_{f_i}$ such that pullbacks of $L$ to $Spec(R_{f_i}((h)))$ are trivial. I don't see why your ring provides a counterexample, so I will be grateful if you explain it in more details. $\endgroup$
    – Ekaterina Bogdanova
    Commented Apr 30, 2020 at 10:39
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    $\begingroup$ You are correct, and I am incorrect. I misread the question. I thought you were just asking about surjectivity of the group homomorphism from $\textbf{GL}(R((h))$ to $\textbf{PGL}(R((h))$. Now I see that is not what you are asking. Sorry for the confusion. $\endgroup$
    – Jason Starr
    Commented May 1, 2020 at 14:50

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